2.1 Ultrafast solid-state lasers - ETH - the Keller Group

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100 2.1.5 Pulse propagation in dispersive media [Ref. p. 134 For example, at a fused quartz prism spacing of 40 cm, the total dispersion that is produced by a double pass through the prism pairs amounts to −862 fs 2 and a third-order dispersion of −970 fs 3 at a center wavelength of 800 nm, assuming zero prism insertion into the beam (Table 2.1.9). We can reduce the negative dispersion by moving the prisms into the laser beam: Each additional millimeter of prism insertion produces a positive dispersion of 101 fs 2 but only a third-order dispersion of 78 fs 3 per cavity round trip. Thus, the prism pairs can generally only be used to compensate for either second- or third-order dispersion. With fused quartz prisms alone pulses as short as 10 fs can be produced with a Ti:sapphire laser [93Asa, 96Flu1]. Slightly shorter pulses can be achieved if the Ti:sapphire laser is operated at a center wavelength of approximately 850 nm, where nearzero second- and third-order dispersion can be obtained for a Ti:sapphire crystal of about 2 mm thickness. In this case 8.5 fs have been generated [94Zho]. Another dispersion compensation is then required for shorter pulse generation. Therefore, the chirped mirrors were designed to show the inverse higher-order group delay dispersion of the dispersion of the prism pair plus laser crystal to eliminate the higher-order dispersion and obtain a slightly negative but constant group delay dispersion required for an ideal soliton pulse. 2.1.5.2.4 Chirped mirrors Dielectric Bragg mirrors with regular quarter-wave layer stacks have a fairly small dispersion when operated well within their reflection bandwidth, but increasing dispersion at the edges of this range. Modified designs can be used to obtain well-controlled dispersion over a large wavelength range. One possibility already discussed is the use of a GTI structure (see Sect. 2.1.5.2.1). Another broad range of designs is based on the concept of the chirped mirrors [94Szi] (Fig. 2.1.16d): If the Bragg wavelength is appropriately varied within a Bragg mirror design, longer wavelengths can be reflected deeper in the mirror structure, thus acquire a larger phase change, which leads to negative dispersion. However, the straight-forward implementation of this idea leads to strong oscillations of the GDD (as a function of frequency) which render such simple designs useless. These oscillations can be greatly reduced by numerical optimizations which introduce complicated (and not analytically explainable) deviations from the simple chirp law. A great difficulty is that the figure of merit to optimize is a complicated function of many layer thickness variables, which typically has a large number of local maxima and minima and thus is quite difficult to optimize. Nevertheless, refined computing algorithms have lead to designs with respectable performance, which were realized with precision growth of dielectric mirrors. Such mirrors can compensate the dispersion in Ti 3+ :sapphire lasers for operation with pulse durations well below 10 fs. Figure 2.1.16d shows a typical chirped mirror structure which schematically shows the path of a long-wavelength (i.e. 1000 nm) and short-wavelength (i.e. 650 nm) beam. This results in negative dispersion, because the long wavelengths are made to penetrate deeper into the mirror structure than short wavelengths. Figure 2.1.16d also shows the standing-wave electric field patterns in a chirped mirror structure versus wavelength. The negative dispersion of the mirrors is clearly illustrated by the dependence of penetration depth on wavelength for the range of 650 to 1050 nm. The highly transmissive region around 500 nm is used for pumping the Ti:sapphire laser through these chirped mirrors. According to [94Szi], chirping means that the Bragg wavelength λ B is gradually increased along the mirror, producing a negative Group Delay Dispersion (GDD). However, no analytical explanation of the unwanted oscillations typically observed in the group delay and GDD of such a simple-chirped mirror was given. These oscillations were minimized purely by computer optimization. The original chirped mirror design was further refined by the Double-Chirped Mirror (DCM) [97Kae, 99Mat] concept which takes into account the impedance matching problem which occurs at the air–mirror interface and the grating structure in the mirror. An exact coupled-mode analysis [97Mat] was used to develop a double-chirp technique. The impedance matching concept allowed a much better insight to the design limitations and allowed for the first time for an analytical design with custom-tailored dispersion characteristics which required only minor numerical optimization Landolt-Börnstein New Series VIII/1B1
Ref. p. 134] 2.1 Ultrafast solid-state lasers 101 [99Mat]. The strong periodic variation in the group delay of the original chirped mirrors occurs due to impedance mismatch between the incident medium (i.e. typically air) and the mirror stack and also within the mirror stack. Using the accurate analytical expressions for phase, group delay, and GDD [99Mat], DCMs could be designed and fabricated with a smooth and custom-tailored GDD suitable for generating pulses in the two-cycle regime directly from a Ti:sapphire laser [99Sut, 99Mor1, 99Mor2]. A Double-Chirped Mirror (DCM) [97Kae, 99Mat] is a multilayer interference coating that can be considered as a composition of at least two sections, each with a different task. The layer materials are typically SiO 2 and TiO 2 . The first section is the AR coating, typically composed of 10 to 14 layers. It is necessary because the theory is derived assuming an ideal matching to air. The other section represents the actual DCM structure, as derived from theory. The double-chirp section is responsible for the elimination of the oscillations in the GDD from within the mirror stack. Double-chirping means that in addition to the local Bragg wavelength λ B the local coupling of the incident wave to the reflected wave is independently chirped as well. The local coupling is adjusted by slowly increasing the high-index layer thickness in every pair so that the total optical thickness remains λ B /2. This corresponds to an adiabatic matching of the impedance. The AR-coating, together with the rest of the mirror, is used as a starting design for a numerical optimization program. Since theoretical design is close to the desired design goal, a local optimization using a standard gradient algorithm is sufficient. At this point, only the broadband AR-coating sets a limitation on the reduction of the GDD oscillations. An AR-coating with a residual reflectivity of less than 10 −4 is required for a DCM at a center wavelength of around 800 nm, which results in a bandwidth of only 250 nm [00Mat]. This bandwidth limitation cannot be removed with more layers in the mirror structure [96Dob]. The invention of the BAck-SIde-Coated (BASIC) mirrors [00Mat] or later the tilted front-side mirrors [01Tem] resolved this issue. In the BASIC mirror the ideal DCM structure is matched to the low-index material of the mirror which ideally matches the mirror substrate material. This DCM structure is deposited on the back of the substrate and the AR-coating is deposited on the front of the slightly wedged or curved substrate, so that the residual reflection is directed out of the beam and does not deteriorate the dispersion properties of the DCM structure on the other side of the substrate. Thus, the purpose of the AR-coating is only to reduce the insertion losses of the mirror at the air–substrate interface. For most applications it is sufficient to get this losses as low as 0.5 %. Therefore, the bandwidth of such an AR-coating can be much broader. Both the DCM and AR-coating multilayer structures can be independently designed and optimized. Based on this analysis, we designed a BAck-SIde-Coated Double-Chirped Mirror (BASIC DCM) that supports a bandwidth of 220 THz with group delay dispersion oscillations of about 2 fs 2 (rms), an order-ofmagnitude improvement compared to previous designs of similar bandwidth [00Mat]. Ultrabroad BASIC DCMs with a bandwidth of 270 THz have also been used to compress supercontinuum of cascaded hollow fibers down to 4.6 fs [04San]. The trade-off is that the substrate has to be as thin as possible to minimize the overall material dispersion. In addition, the wedged mirror leads to an undesired angular dispersion of the beam. Another possibility to overcome the AR-coating problem is given with the idea to use an ideal DCM under Brewster-angle incidence [03Ste]. In this case, the low-index layer is matched to air. However, under p-polarized incidence the index contrast and therefore the Fresnel reflectivity of a layer pair is reduced and more layer pairs are necessary to achieve high reflectivity. This increases the penetration depth into the mirror which has the advantage that these mirrors can produce more dispersion per reflection but this means that scattering and other losses and also fabrication tolerances become even more severe. In addition, this concept is difficult to apply to curved mirrors. Furthermore, the spatial chirp of the reflected beam has to be removed by back reflection or an additional reflection from another Brewster-angle mirror. Other methods to overcome the AR-coating problem are based on using different chirped mirrors with slightly shifted GTI oscillations that partially cancel each other. Normally, these chirped mirrors are very difficult to fabricate [00Mat]. Many different growth runs normally result in strong Landolt-Börnstein New Series VIII/1B1
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2.1 Ultrafast solid-state lasers - ETH - the Keller Group

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